Chelsea is completely different from other elite EPL club. They're strict when it comes to their managers and not welcoming any relenting outcomes. Erik Ten Hag in Chelsea? He would have been sacked long time ago. The previous owner, Abramovic doesn't welcome any relenting efforts or if he sensed any form of incompetence from the headcoach, he will definitely make good use of the sack letter.
Going by that which you have just mentioned about Chelsea, pochettino doesn't fit into any of the displeasure you have expressed in this post but was still sacked so I don't think it's a problem with the coach it's a problem with the management not been patient enough because there are Coaches who may not do so well in their first season but will be the best they will have in subsequent seasons.
The case of ten hag, he has been given what feels like morethan enough time yet he was unable to deliver and he just keep using some kind of light weighted victory to creat relevance by saying he won a trophy a d all that meanwhile he's performance is just too inconsistent that he's not productive,r the trophies he has won in the last two seasons are such that feels like consolation and not the type the team will celebrate well enough as them been able to have won something really great for the season. There are trophies you win a d it sounds like result of hard work and there are others that everyone see as something you could just win.
FA cup actually isn't really light weighted but more were expected of ten hag and if the management of Manchester united still decides to keep him then for whatever reason it may be I think it's rather going to be more detrimental than beneficial except they will seel of some players aswell and bring in new ones as recommended by ten hag which will be seen as those he can work with.